Geodesic
A group theory derivation of the relativistic path integral and the “history-string” dynamics
Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 3, pp. 392-415 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We derive the Feynman path integral for relativistic elementary particles using group theory considerations. We apply an approach in which choosing a symmetry group (or semigroup) allows deriving the kinematics and dynamics of a particle including the state space and the propagator from it. The quantum properties of a particle appear from intertwining two representations of the symmetry (semi)group, one of which describes local properties of the particle and the other describes the particle as a whole. The path-integral-like dynamics appears when the symmetry semigroup has a structure similar to that of the relativistic analogue of the Galilei group (in which the Lorentz-invariant “proper time” plays the role of time) with translations replaced with the semigroup of trajectories (parameterized paths). The classical action in the weight functional of the path integral is defined by this semigroup up to couplings to gauge and/or gravitational fields. The obtained formalism is suitable for describing not only pointlike particles but also nonlocal objects of the “history-string” type, which, as previously shown, allow explaining quark confinement.
Keywords: path integral, symmetry, path group, path-dependent function, quantum string. 
@article{TMF_2012_173_3_a3,
     author = {M. B. Menskii},
     title = {A~group theory derivation of the~relativistic path integral and the~{\textquotedblleft}history-string{\textquotedblright} dynamics},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {392--415},
     year = {2012},
     volume = {173},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_173_3_a3/}
}
TY  - JOUR
AU  - M. B. Menskii
TI  - A group theory derivation of the relativistic path integral and the “history-string” dynamics
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2012
SP  - 392
EP  - 415
VL  - 173
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2012_173_3_a3/
LA  - ru
ID  - TMF_2012_173_3_a3
ER  - 
%0 Journal Article
%A M. B. Menskii
%T A group theory derivation of the relativistic path integral and the “history-string” dynamics
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2012
%P 392-415
%V 173
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2012_173_3_a3/
%G ru
%F TMF_2012_173_3_a3
M. B. Menskii. A group theory derivation of the relativistic path integral and the “history-string” dynamics. Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 3, pp. 392-415. http://geodesic.mathdoc.fr/item/TMF_2012_173_3_a3/

[1] E. Wigner, Ann. Math., 40:1 (1939), 149–204 | DOI | MR | Zbl

[2] T. Newton, E. Wigner, Rev. Modern Phys., 21:3 (1949), 400–406 | DOI | Zbl

[3] A. S. Wightman, Rev. Modern Phys., 34:4 (1962), 845–872 | DOI | MR

[4] M. B. Mensky, Commun. Math. Phys., 47:2 (1976), 97–108 | DOI | MR | Zbl

[5] M. B. Menskii, Metod indutsirovannykh predstavlenii: prostranstvo-vremya i kontseptsiya chastits, Nauka, M., 1976

[6] M. B. Menskii, TMF, 57:2 (1983), 217–231 | DOI | MR

[7] S. Mandelstam, Ann. Phys., 19:1 (1962), 1–24 ; 25–66 | DOI | MR | Zbl | DOI | MR | Zbl

[8] S. Mandelstam, Phys. Rev., 175:5 (1968), 1580–1603 ; 1604–1623 | DOI | DOI

[9] M. B. Menskii, TMF, 18:2 (1974), 190–202 | DOI | MR

[10] M. B. Menskii, Gruppa putei: izmereniya, polya, chastitsy, Nauka, M., 1983 | MR

[11] M. Süveges, Acta Phys. Acad. Sci. Hung., 20 (1966), 41–50 ; 20 (1966), 51–58 ; 22 (1966), 273–284 ; 27 (1969), 261–268 | DOI | MR | DOI | MR | DOI | MR | DOI | MR

[12] M. B. Mensky, Lett. Math. Phys., 2:3 (1978), 175–180 | DOI | MR

[13] M. B. Mensky, Lett. Math. Phys., 3:6 (1979), 513–520 | DOI | MR

[14] M. B. Mensky, Helv. Phys. Acta, 69:3 (1996), 301–304, arXiv: gr-qc/0010105 | MR | Zbl

[15] J. J. Aghassi, P. Roman, R. M. Santilli, Phys. Rev. D, 1:10 (1970), 2753–2765 | DOI | Zbl

[16] E. C. Stueckelberg, Helv. Phys. Acta, 14 (1941), 321–322 ; 15 (1942), 23–37 | MR | Zbl | MR | Zbl

[17] J. J. Aghassi, P. Roman, R. M. Santilli, J. Math. Phys., 11 (1970), 2297–2301 | DOI | MR

[18] J. J. Aghassi, P. Roman, R. M. Santilli, Nuovo Cimento A, 5:4 (1971), 551–590 | DOI | MR

[19] M. B. Menskii, ZhETF, 90:2 (1986), 416–428 | MR